Ambient locally finite mod-two relative chains for candidate regions

Fix an ambient locally finite chain $c$. In a neighborhood meeting only finitely many of its simplex images, every contributing face comes from one of that finite list and each parent has finitely many faces. Consequently every coefficient of $\mathrm{\partial }c$ is a finite mod-two sum and the resulting chain is locally finite. Applying the boundary twice near any point therefore reduces to finitely many parents; [F3] cancels their codimension-two faces in pairs. Thus ${\mathrm{\partial }}^{2}c=0$. Every face of a simplex mapping into $B$ still maps into $B$, so the $B$-supported groups form a subcomplex. Hence $[c]\mapsto [\mathrm{\partial }c]$ is well-defined on the quotient, squares to zero and supplies the displayed homology groups, by the same quotient argument as [F2].

Relative cycles are exactly chains with boundary supported in $B$. In the quotient, $[a]=[x]$ in homology means $a+x$ is a quotient boundary, which says exactly that $a+x=\mathrm{\partial }r+b$ for some ambient locally finite $r$ and some $B$-supported $b$. Over ${\mathbb{Z}}_2$ this is the displayed condition. Adding quotient boundaries to $a$ or $x$ merely adds the corresponding chains to $r$; adding $B$-supported chains changes only $b$, proving representative invariance. If $\mathrm{\Sigma }$ is compact, finitely many of the neighborhoods in the local-finiteness definition cover it. Their finite union meets every nonzero simplex, so only finitely many nonzero simplex images occur. Thus the ambient groups, their $B$-supported subgroups, boundaries and quotient homology are the ordinary finite ones in [F2]. Every finite chain remains locally finite even when $\mathrm{\Sigma }$ is not compact; no assertion that the two homology theories then coincide is needed.

For the fundamental-chain local condition, ambient local finiteness makes the list of simplices containing a fixed $p$ finite. Since all their images lie in the embedded subspace $Y$, they are singular simplices of $Y$ itself. Any face whose image contains $p$ comes from a parent containing $p$. Its coefficient in $\mathrm{\partial }{c}_p$ therefore equals its coefficient in $\mathrm{\partial }c$. When $p$ is an interior point of $Y$, the latter boundary has no such face, by the stated boundary-support condition. Thus $\mathrm{\partial }{c}_p$ is a finite chain in $Y\setminus \{p\}$, and its ordinary point-local relative class is defined. By [F4] the prescribed nonzero element is unambiguous over ${\mathbb{Z}}_2$, regardless of orientability. This establishes a precise, internally consistent local test for the supplied chain data; it does not assume their existence.

In the sufficient triangulated construction, parametrize every simplex by listing its vertices in the supplied increasing order. A common face then has literally the same singular parametrization from either incident top simplex. The ambient local-finiteness hypothesis makes the top-simplex sum a chain in $C_n^{\mathrm{l}\mathrm{f}}(\mathrm{\Sigma })$. Interior faces occur twice and cancel modulo two, while boundary faces occur once, so its boundary is supported in $\mathrm{\partial }Y$. Proper inclusion is sufficient for the required ambient local finiteness by the compact-neighborhood argument in [F1]; it is not imposed on other supplied chains in the general branch. For an interior point $p$, the finite compact certificate $N_p$ contains a neighborhood of $p$. Local finiteness ensures that, after shrinking that neighborhood, no top simplex outside its finite subcomplex meets it: among the finitely many nearby closed simplex images, none outside that subcomplex can contain $p$, because its interior would then approach $p$ from outside the neighborhood contained in $N_p$. Hence the point-local chain obtained from the global sum is exactly the point-local chain of the compact certificate. Its class in $H_n(N_p,N_p\setminus \{p\};{\mathbb{Z}}_2)$ is nonzero by the supplied compact relative fundamental-class property and [F4]. Set $Z:=Y\setminus N_p$. Since $N_p$ contains a neighborhood of $p$, $p\mathrm{\notin }\bar{Z}$; since $Y$ is Hausdorff, $Y\setminus \{p\}$ is open. Thus $\bar{Z}\subseteq {\mathrm{int}}_Y(Y\setminus \{p\})$. By [F6], inclusion $(N_p,N_p\setminus \{p\})\to (Y,Y\setminus \{p\})$ induces an isomorphism. It sends the certificate class to the global point-local class, which is therefore nonzero. Step 2.2 therefore certifies the entire global top chain as fundamental. For an $n$-dimensional region with $\mathrm{\partial }r=a+x+b$, $b$ supported in $B$, and $\mathrm{\partial }a,\mathrm{\partial }x$ supported in $B$, face cancellation supplies the witness and $a,x$ are relative cycles. Step 2.1 then gives $[a]=[x]$. This proves the sufficient construction without a universal triangulation or candidate-existence theorem.